Chapter 3 Geometry of convex functions - Meboo Publishing ...

3.8. MATRIX-VALUED CONVEX FUNCTION 257
3.8.3 second-order condition, matrix function
The following line theorem is a potent tool for establishing convexity of a
multidimensional function. To understand it, what is meant by line must first
be solidified. Given a function g(X) : R p×k →S M and particular X, Y ∈ R p×k
not necessarily in that function’s domain, then we say a line {X+ t Y | t ∈ R}
passes through domg when X+ t Y ∈ domg over some interval of t ∈ R .
3.8.3.0.1 Theorem. Line theorem. [59,3.1.1]
Matrix-valued function g(X) : R p×k →S M is convex in X if and only if it
remains convex on the intersection of any line with its domain. ⋄
Now we assume a twice differentiable function.
3.8.3.0.2 Definition. Differentiable convex matrix-valued function.
Matrix-valued function g(X) : R p×k →S M is convex in X iff domg is an
open convex set, and its second derivative g ′′ (X+ t Y ) : R→S M is positive
semidefinite on each point of intersection along every line {X+ t Y | t ∈ R}
that intersects domg ; id est, iff for each and every X, Y ∈ R p×k such that
X+ t Y ∈ domg over some open interval of t ∈ R
Similarly, if
d 2
dt 2 g(X+ t Y ) ≽ S M +
0 (611)
d 2
dt g(X+ t Y ) ≻ 0 (612)
2
S M +
then g is strictly convex; the converse is generally false. [59,3.1.4] 3.21 △
3.8.3.0.3 Example. Matrix inverse. (confer3.4.1)
The matrix-valued function X µ is convex on int S M + for −1≤µ≤0
or 1≤µ≤2 and concave for 0≤µ≤1. [59,3.6.2] In particular, the
function g(X) = X −1 is convex on int S M + . For each and every Y ∈ S M
(D.2.1,A.3.1.0.5)
d 2
dt 2 g(X+ t Y ) = 2(X+ t Y )−1 Y (X+ t Y ) −1 Y (X+ t Y ) −1 ≽
3.21 The strict-case converse is reliably true for quadratic forms.
S M +
0 (613)